You can connect as many statements as you want together to form a compound
statement. Two compound statements are equivalent if they have the same
truth tables.

2.3.2.1 Implication

Let p = "You are beautiful", and let q = "I will be happy",
then
p >
q (p implies q) means "If you are beautiful, then I will be happy".
Now, what is the truth table for p >
q?

Now, if you ARE beautiful, AND I AM happy, then p Þ
q is obviously true. But what if you are not beautiful, and I am happy?
Well, the only thing I have stated is that if you are beautiful, then
I will be happy. I haven't said I won't be happy if you aren't beautiful,
now have I? I might be happy for other reasons. Hence, if you aren't beautiful,
but I am still happy, p >
q is still true! What if you are not beautiful, and I am not happy? Have
I lied? No, I haven't said anything about what happens if you are not
beautiful, so p Þ
q is still true. Last case: what if you are beautiful, and I am not happy?
Well, then I have lied, so p >
q is false. To sum it up:

p

q

p >
q

T

T

T

F

T

T

F

F

T

T

F

F

Notice that p >
q is true when [(p is false) OR (q is true)], which means that p >
q =
~p v
q

Also notice that if false = 0 and true = 1, then p implies q (p >
q) can be calculated with normal mathetmatical operators as follows:

Sometimes, p is called the hypothesis, and q is called the conclusion.
From the truth table (and our little discussion), we se that an implication
is false only when the hypothesis is true and the conclusion is false.